Statistical stability of three and more body hierarchical systems in celestial mechanics
itqquad Then to the Heav'n itself I cried,qquad Asking, `What Lamp had destiny to guideqquad Her little Children stumbling in the Dark?'qquad And - `A blind understandingMissing data. Heav'n replied. qquadqquadqquad Rubaiyat of Omar Khayyam It seems that not everybody in Persia in the eleventh century was as convinced as the astrologers that the movements of the heavens controlled the destiny of Man. Nevertheless, for many centuries before and since, kings and emperors rewarded handsomely those astronomer/astrologers who could give them advice based on the movements of the planets and other celestial bodies. (There may be some astronomers today who would wish for similar generous patronage). Since the advent of modern celestial mechanics with the work of Isaac Newton, orbital motion has been studied for its own sake, and in the last thirty years, for the purposes of sending artificial satellites and manned craft into space. Yet for 300 years, one of the most important questions posed by celestial mechanics remains unanswered: are the motions of the planets in the Solar System stable? Could planets collide or even escape? Countless workers since Newton's time have sought Lamps to the destiny of the Solar System, but our Understanding is still obscured by many blind-spots. This thesis does not claim to give any definitive answers to these questions. It does indicate how to obtain quantitative estimates of the likelihood of certain events occurring. Simple statistical methods are applied to the results of numerical experiments and give probabilities of planetary orbits crossing or bodies escaping dynamical systems altogether. In Chapter 1 a general review of the problem of the Solar System's stability is given along with brief descriptions of methods and definitions of stability which have been used in the past. This thesis studies the stability of real and fictitious dynamical systems not necessarily associated with the Solar System. It investigates one particular definition of stability, namely hierarchical stability, using special perturbation methods. The definitions of hierarchical systems, hierarchical stability and empirical stability parameters are reviewed in Chapter 2. These will form the basis for subsequent numerical experiments. One further definition of stability - Hill stability is an important condition for hierarchical stability. It has been studied in a mathematically rigorous way in the problem of three massive bodies in mutually perturbed orbits. This analysis as well as some new numerical results are given in Chapter 3. Numerical integration experiments were carried out, with the aid of a mainframe computer, to study the period of time for which various three-body systems remain stable. Several hundred fictitious systems with different masses and starting conditions were studied. In each case, all three bodies' orbits lay in the same plane. In some systems, all the bodies orbited in the same direction (direct); for other systems, one body orbited in the opposite direction from the other two (retrograde). The results of these experiments are presented in Chapter 4 (for retrograde systems) and Chapter 5 (for direct systems). The results are grouped in such a way that analytical curves may be fitted to the data. This allows predictions of stability lifetimes for similar systems without the need for lengthy numerical integration experiments. Systems whose masses, initial positions and initial velocities fall into certain ranges are always stable. These regions of hierarchical stability are mapped out and compared with corresponding regions of Hill stability. In the case of direct systems, commensurabilities give rise to large fluctuations in stability lifetimes, if the initial conditions are varied slightly. Additional statistical methods are described in Chapter 5 to cope with this effect. In Chapter 6, the results of Chapters 4 and 5 are compared with real three-body systems within the Solar System. Possible origins of the Solar System are discussed in the light of the results.